We show how dipolar interactions between dysprosium atoms in an optical
lattice can be used to obtain fractional quantum Hall states. In our approach,
dysprosium atoms are trapped one atom per site in a deep optical lattice with
negligible tunneling. Microwave and spatially dependent optical dressing fields
are used to define an effective spin-1/2 or spin-1 degree of freedom in each
atom. Thinking of spin-1/2 particles as hardcore bosons, dipole-dipole
interactions give rise to boson hopping, topological flat bands with Chern
number 1, and the \nu = 1/2 Laughlin state. Thinking of spin-1 particles as
two-component hardcore bosons, dipole-dipole interactions again give rise to
boson hopping, topological flat bands with Chern number 2, and the bilayer
Halperin (2,2,1) state. By adjusting the optical fields, we find a phase
diagram, in which the (2,2,1) state competes with superfluidity.
Generalizations to solid-state magnetic dipoles are discussed.
